Access token gets invalidated after logging in on a different computer

Using token-based login with one Minecraft account on two computers has become impossible. As soon as the second client logs in, the first's access token gets invalidated, prompting the user for the password on the next launch.

Steps to reproduce:

• Start client 1 on computer 1
• Log in on client 1
• Close client 1
• Start client 2 on computer 2
• Log in on client 2 using the same account as on client 1
• Close client 2
• Start client 1 on computer 1